Metasurfaces as optically thin composite layers can be modeled as electric and magnetic surface current sheets flowing in the layer volume in the metasurface plane. In the most general linear metasurface, the electric surface current can be induced by both incident electric and magnetic fields. Likewise, magnetic polarization and magnetic current can be induced also by external electric field. Metasurfaces which exhibit magnetoelectric coupling are called bianisotropic metasurfaces. In this review, we explain the role of bianisotropic properties in realizing various metasurface devices and overview the state-of-the-art of research in this field. Interestingly, engineered bianisotropic response is seen to be required for realization of many key field transformations, such as anomalous refraction, asymmetric reflection, polarization transformation, isolation, and more. Moreover, we summarize previously reported findings on uniform and gradient bianisotropic metasurfaces and envision novel and prospective research directions in this field.